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Lecture 12 Finitely Repeated Games
14.12 Game Theory Muhamet Yildiz
1
Road Map
- 1. Entry-Deterrence/Chain-store paradox
- 2. Finitely repeated Prisoners Dilemma
- 3. A general result
- 4. Repeated games with multiple equilibria
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Finitely Repeated Games 14.12 Game Theory Muhamet Yildiz 1 Road - - PDF document
Finitely Repeated Games 14.12 Game Theory Muhamet Yildiz 1 Road - - PDF document
Lecture 12 Finitely Repeated Games 14.12 Game Theory Muhamet Yildiz 1 Road Map 1. Entry-Deterrence/Chain-store paradox 2. Finitely repeated Prisoners Dilemma 3. A general result 4. Repeated games with multiple equilibria 2 Prisoners' Dilemma,
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Prisoners' Dilemma, repeated twice, many times
- Two dates T = {O,I};
- At each date the prisoners' dilemma is played:
C
D
C 5,5 0,6 D 6,0
1,1
- At the beginning of I players observe the strategies at o.
Payoffs= sum of stage payoffs. 3
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Twice-repeated PD
I
c
- 2
c
D
c
D 1 1
- 105116
5061
11
6 12 7 6 1 7 2 10
11
5 6
11
12 6 7
56016712
What would happen ifT = {O, I ,2, ... ,n}?
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Microsoft v. a Startup
Startup NP
SE
u
NU
u
- 2
1 1 1 2
3 4 5
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Microsoft v. Startups
su
NP
SE
M S
u
U / MS
sur
SU M U NUj::
NUj
MS M
NU
U NUj::
NUj :
0 2
2 4
3 3
1 3 2 2
3
2 2 2 3
4 5 3 4 5 6 4 5 6 7 5 6 7 8 What would happen ifthere are n startups? 6
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Entry deterrence
1 Enter 2 Acc.
... ...
(1 ,1)
X
Fight
,
,
,
,
(0,2) (-1,-1)
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Entry deterrence, repeated twice
1 Frier 2 A::c.
\ Frier
2 A::c.
~
(2,2)
:x Fil#: :x HI#:
A::c.
2 Frier
~
\,3) ~
(1,3)
(0,0)
~
Frier 2 A::c.
HI#: :x
(0,0)
:x Fil#: (-1,1 )
(0,4) (-\,\) (-2,-2)
8
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A general result
- G = "stage game" = a finite game
- T = {O,l, ...
,n}
- At each t in T, G is played, and players remember
which actions taken before t;
- Payoffs = Sum of
payoffs in the stage game.
- Call this game G(T).
Theorem: If G has a unique subgame-perfect equilibrium s*, G(T) has a unique subgame- perfect equilibrium, in which s* is played at each stage.
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With multiple equilibria
T={O,l}
s* =
- At t = 0, play (B,M)
2
I
L M R
- At t = I, play (C,R) if (B,M) at t
= 0, play (A,L) otherwise.
A
1,1 5,0 0,0
B
0,5
4,4
0,0
L M
R
C
0,0 0,0
3,3
A
2,2
6,1
1,1
B 1,6
7,7
1,1
C
1,1
1,1
4,4
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Can you see on the path ofSPE?
T={O,l}
L M R
- (B,M) (B,M)
A
1,1
5,0 0,0
- (B,M) (A,L)
B
0,5
4,4
0,0
- (B,L) (C,R)
C
0,0 0,0
3,3
- (C,L)(C,R)
- Take T={0,1 ,2}
- (C,L) (B,M) (C,R)
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